Infinite quotient rule?

Calculus Level 3

\[ F(x) = \cfrac{x}{x + \cfrac{x}{x + \cfrac{x}{x + \cfrac{x }{._{._.}}} } } \]

If the value of first derivative of \(F(x) \) at \(x = 1\) can be expressed as \( \dfrac{a\sqrt b - b}{c} \) for integers \(a,b\) and \(c\), find the value of \(a+b+c\).